Inversion relations, reciprocity and polyominoes

Bousquet-Mélou, Mireille; Guttmann, A J; Orrick, William P.; Rechnitzer, Andrew

doi:10.1007/bf01608785

Cited by 11 publications

(31 citation statements)

References 24 publications

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“…All (4, 3)-polycycles are: (4 3 ), (4 3 )-v, (4 3 )-e, P 2 × P n for any natural n and two infinite ones:…”

Section: Proper Polycycles Versus Helicenesmentioning

confidence: 99%

“…(i) Any (r, q)-polycycle with (r, q) = (3, 3), (3,4), (4,3), (3,5), (5, 3) is a union of elementary polycycles without common faces,…”

Section: Lemmamentioning

confidence: 99%

“…(ii) for (r, q) = (6, 3), (4,4), (3,6) and for the case r ≥ 7, q ≥ 3, there is a continuum of connected (components of ) kernels of (r, q)-polycycles.…”

Section: Lemmamentioning

confidence: 99%

“…The number of such polycycles, which are not extendible without loosing this property, is equal to 0 for (p, q) = (3, 3), (3,4) and equal to 1 for (p, q) = (4, 3) (it is P 2 × P 5 ). This number is finite for (p, q) = (5, 3), (3,5) and infinite, otherwise.…”

Section: Non-extendible Polycyclesmentioning

confidence: 99%

“…this polyhedral sphere permits to consider the cross-ring as "not simply-connected (8, 3)-polycycle" with p 8 = 5; this polyhedron have five regular 8-gonal faces: four indicated by the number "8" on the left-hand side of Figure 8 and one bounded by C 11,12,4,5,25,26,18,19 . The handle prevents it from embedding into R 2 and from cell-homomorphism into (8 3 ).…”

Section: Theorem 11mentioning

confidence: 99%

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Clusters of cycles

Deza¹,

Штогрин²

2002

Journal of Geometry and Physics

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A cluster of cycles (or (r, q)-polycycle) is a simple planar 2-connected finite or countable graph G of girth r and maximal vertex-degree q, which admits an (r, q)-polycyclic realization P (G) on the plane. An (r, q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r ) are combinatorial r-gons, (iii) all vertices, edges and interior faces form a cell-complex.An example of (r, q)-polycycle is the skeleton of (r q ), i.e. of the q-valent partition of the sphere, Euclidean plane or hyperbolic plane by regular r-gons. Call spheric pairs (r, q) = (3, 3), (4, 3), (3, 4), (5, 3), (3, 5). Only for those five pairs, P ((r q )) is (r q ) without exterior face; otherwise, P ((r q )) = (r q ).Here we give a compact survey of results on (r, q)-polycycles. We start with the following general results for any (r, q)-polycycle G: (i) P (G) is unique, except of (easy) case when G is the skeleton of one of 5 Platonic polyhedra; (ii) P (G) admits a cell-homomorphism f into (r q ); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented.Call a polycycle proper if it is a partial subgraph of (r q ) and a helicene, otherwise. In [18] all proper spheric polycycles are given. An (r, q)-helicene exists if and only if p r > (q − 2)(r − 1) and (r, q) = (3, 3). We list the (4, 3)-, (3, 4)-helicenes and the number of (5, 3)-, (3, 5)-helicenes for first interesting p r . Any outerplanar (r, q)-polycycle G is a proper (r, 2q − 2)-polycycle and its projection f (P (G)) into (r 2q−2 ) is convex. Any outerplanar (3, q)-polycycle G is a proper (3, q + 2)-polycycle.

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“…All (4, 3)-polycycles are: (4 3 ), (4 3 )-v, (4 3 )-e, P 2 × P n for any natural n and two infinite ones:…”

Section: Proper Polycycles Versus Helicenesmentioning

confidence: 99%

“…(i) Any (r, q)-polycycle with (r, q) = (3, 3), (3,4), (4,3), (3,5), (5, 3) is a union of elementary polycycles without common faces,…”

Section: Lemmamentioning

confidence: 99%

“…(ii) for (r, q) = (6, 3), (4,4), (3,6) and for the case r ≥ 7, q ≥ 3, there is a continuum of connected (components of ) kernels of (r, q)-polycycles.…”

Section: Lemmamentioning

confidence: 99%

Section: Non-extendible Polycyclesmentioning

confidence: 99%

Section: Theorem 11mentioning

confidence: 99%

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